Working with Unlabeled Data Clustering Analysis. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

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1 Working with Unlabeled Data Clustering Analysis Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

2 Unsupervised learning Finding centers of similarity using the popular k-means algorithm Taking a bottom-up approach to building hierarchical clustering trees Identifying arbitrary shapes of objects using a densitybased clustering approach 2

3 K-means clustering 3

4 K-means clustering 1. Randomly pick k centroids from the sample points as initial cluster centers. 2. Assign each sample to the nearest centroid μ (j), j=1 k. 3. Move the centroids to the center of the samples that were assigned to it. 4. Repeat steps 2 and 3 until the cluster assignments do not change or a user-defined tolerance or maximum number of iterations is reached. 4

5 Similarity measures between objects Squared Euclidean distance between two points x and y in m-dimensional space 5

6 Optimization by minimizing cluster inertia Iterative approach for minimizing the within-cluster Sum of Squared Errors (SSE) μ (j) is the representative point (centroid) for cluster j w (i, j) = 1 if the sample x (i) is in cluster j w (i, j) = 0 otherwise. 6

7 A smarter way of placing the initial cluster centroids using k-means++ 1. Initialize an empty set M to store the k centroids being selected. 2. Randomly choose the first centroid μ (j) from the input samples and assign it to M. 3. For each sample x (i) that is not in M, find the minimum squared distance d(x (i),m) to any of the centroids in M. 4. To randomly select the next centroid μ (p), use a weighted probability distribution equal to 5. Repeat steps 2 and 3 until k centroids are chosen. 6. Proceed with the classic k-means algorithm. 7

8 Fuzzy C-means (FCM) Cluster membership vector of a sample x in k-means Cluster membership vector in FCM 8

9 FCM algorithm 1. Specify the number of k centroids and randomly assign the cluster memberships for each point. 2. Compute the cluster centroids μ (j), j=1 k. 3. Update the cluster memberships for each point. 4. Repeat steps 2 and 3 until the membership coefficients do not change, or a user-defined tolerance or maximum number of iterations is reached. 9

10 Cluster membership probability The membership of the x (i) sample belonging to the u (j) cluster Typically m = 2, so-called fuzziness coefficient 10

11 Cluster centroid The center u (j) of cluster j is calculated as the mean of all n samples weighted by the degrees to which each sample belongs to that cluster w m(i,j) 11

12 The objective function of FCM Similar to the within cluster sum-squared-error for minimization in k-means 12

13 The elbow method to find the optimal number of clusters Within-cluster SSE (distortion) can be used to quantify the quality of clustering 13

14 Quantifying the quality of clustering via silhouette plots Plot a measure of how tightly grouped the samples in the clusters are 1. Calculate the cluster cohesion a (i) as the average distance between a sample x (i) and all other points in the same cluster. 2. Calculate the cluster separation b (i) from the next closest cluster as the average distance between the sample x (i) and all samples in the nearest cluster 3. Calculate the silhouette s (i) 14

15 Silhouette plots 15

16 Hierarchical clustering Agglomerative hierarchical clustering Start with one cluster that encompasses all our samples Iteratively split the cluster into smaller clusters until each cluster only contains one sample. Divisive hierarchical clustering Start with each sample as an individual cluster Merge the closest pairs of clusters until only one cluster remains. 16

17 Agglomerative clustering using complete linkage approach Compute the distance matrix of all samples. Represent each data point as a singleton cluster. Merge the two closest clusters based on the distance between the most dissimilar (distant) members. Update the similarity matrix. Repeat steps 2-4 until one single cluster remains 17

18 Cluster mergence based on complete linkage vs. single linkage 18

19 Distance matrix Pairwise distance Dendrogram 19

20 Density-based spatial clustering of applications with noise (DBSCAN) Density-based clustering assigns cluster labels based on dense regions of points 20

21 Point labeling in DBSCAN algorithm A point is considered a core point if at least a specified number (MinPts) of neighboring points fall within the specified radius A border point is a point that has fewer neighbors than MinPts within but lies within the radius of a core point All other points that are neither core nor border points are considered noise points 21

22 DBSCAN algorithm 1. Form a separate cluster for each core point or connected group of core points Core points are connected if they are no farther away than 2. Assign each border point to the cluster of its corresponding core point. 22

23 Advantages of DBSCAN It does not assume that the clusters have a spherical shape as in k-means. It doesn't necessarily assign each point to a cluster but is capable of removing noise points. 23

24 Reference Sebastian Raschka, Vahid Mirjalili. Python Machine Learning: Machine Learning and Deep Learning with Python, scikit-learn, and TensorFlow. Second Edition. Packt Publishing,